Two-parameter Hardy-Littlewood inequality and its variants

Let s* denote the maximal function associated with the rectangular partial sums ${\displaystyle s_{mn}(x,y)}$ of a given double function series with coefficients ${\displaystyle c_{jk}}$. The following generalized Hardy-Littlewood inequality is investigated: ${\displaystyle {\left||s\cdot |\right|}_{p,\mu }\le C_{p,\alpha ,\beta }{\left\{{\Sigma }_{j=0}^{\infty }{\Sigma }_{k=0}^{\infty }{(j̅)}^{p-\alpha -2}{(k̅)}^{p-\beta -2}{\left|c_{jk}\right|}^p\right\}}^{\frac{1}{p}}}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on ${\displaystyle c_{jk}}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||_{p,μ}-convergence property of ${\displaystyle s_{mn}(x,y)}$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].